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Generalized solutions to nonlinear stochastic differential equations with vector--valued impulsive controls
Generalized exterior sphere conditions and $\varphi$-convexity
1. | Computer Science and Mathematics Division, Lebanese American University, Byblos Campus, P.O. Box 36, Byblos |
2. | Department of Mathematics and Statistics, Concordia University, 1400 De Maisonneuve Blvd. West, Montreal, Quebec H3G 1M8, Canada |
3. | Department of Computer Science and Mathematics, Lebanese American University, Byblos Campus, P.O. Box 36, Byblos, Lebanon |
References:
[1] |
P. Cannarsa and H. Frankowska, Interior sphere property of attainable sets and time optimal control problems, ESAIM: Control Optim. Calc. Var., 12 (2006), 350-370.
doi: doi:10.1051/cocv:2006002. |
[2] |
P. Cannarsa and C. Sinestrari, Convexity properties of the minimum time function, Calc. Var., 3 (1995), 273-298.
doi: doi:10.1007/BF01189393. |
[3] |
P. Cannarsa and C. Sinestrari, "Semiconcave Functions, Hamiton-Jacobi Equations and Optimal Control," Birkhäser, Boston, 2004. |
[4] |
A. Canino, On $p$-convex sets and geodesics, J. Differential Equations, 75 (1988), 118-157.
doi: doi:10.1016/0022-0396(88)90132-5. |
[5] |
F. H. Clarke, "Optimization and Nonsmooth Analysis," Classics in Applied Mathematics, 5, SIAM, Philadelphia, PA, 1990. |
[6] |
F. H. Clarke, Yu. Ledyaev, R. J. Stern and P. R. Wolenski, "Nonsmooth Analysis and Control Theory," Graduate Texts in Mathematics, 178, Springer-Verlag, New York, 1998. |
[7] |
F. H. Clarke and R. J. Stern, State constrained feedback stabilization, SIAM J. Control Optim., 42 (2003), 422-441.
doi: doi:10.1137/S036301290240453X. |
[8] |
F. H. Clarke, R. J. Stern and P. R. Wolenski, Proximal smoothness and the lower-$C^2$ property, J. Convex Anal., 2 (1995), 117-144. |
[9] |
G. Colombo and A. Marigonda, Differentiability properties for a class of non-convex functions, Calc. Var., 25 (2005), 1-31.
doi: doi:10.1007/s00526-005-0352-7. |
[10] |
G. Colombo, A. Marigonda and P. R. Wolenski, Some new regularity properties for the minimal time function, SIAM J. Control Optim., 44 (2006), 2285-2299.
doi: doi:10.1137/050630076. |
[11] |
H. Federer, Curvature measures, Trans. Amer. Math. Soc., 93 (1959), 418-491. |
[12] |
C. Nour, R. J. Stern and J. Takche, Proximal smoothness and the exterior sphere condition, J. Convex Anal., 16 (2009), 501-514. |
[13] |
C. Nour, R. J. Stern and J. Takche, The $\theta$-exterior sphere condition, $\varphi$-convexity, and local semiconcavity, Nonlinear Anal. Theor. Meth. Appl., 73 (2010), 573-589.
doi: doi:10.1016/j.na.2010.04.001. |
[14] |
R. A. Poliquin and R. T. Rockafellar, Prox-regular functions in variational analysis, Trans. Amer. Math. Soc., 348 (1996), 1805-1838.
doi: doi:10.1090/S0002-9947-96-01544-9. |
[15] |
R. A. Poliquin, R. T. Rockafellar and L. Thibault, Local differentiability of distance functions, Trans. Amer. Math. Soc., 352 (2000), 5231-5249.
doi: doi:10.1090/S0002-9947-00-02550-2. |
[16] |
R. T. Rockafellar, Clarke's tangent cones and the boundaries of closed sets in $\R^n$, Nonlinear Anal. Theor. Meth. Appl., 3 (1979), 145-154.
doi: doi:10.1016/0362-546X(79)90044-0. |
[17] |
R. T. Rockafellar and R. J.-B. Wets, "Variational Analysis," Grundlehren der Mathematischen Wissenschaften, 317, Springer-Verlag, Berlin, 1998. |
[18] |
A. S. Shapiro, Existence and differentiability of metric projections in Hilbert spaces, SIAM J. Optim., 4 (1994), 231-259.
doi: doi:10.1137/0804006. |
[19] |
C. Sinestrari, Semiconcavity of the value function for exit time problems with nonsmooth target, Commun. Pure Appl. Anal., 3 (2004), 757-774.
doi: doi:10.3934/cpaa.2004.3.757. |
show all references
References:
[1] |
P. Cannarsa and H. Frankowska, Interior sphere property of attainable sets and time optimal control problems, ESAIM: Control Optim. Calc. Var., 12 (2006), 350-370.
doi: doi:10.1051/cocv:2006002. |
[2] |
P. Cannarsa and C. Sinestrari, Convexity properties of the minimum time function, Calc. Var., 3 (1995), 273-298.
doi: doi:10.1007/BF01189393. |
[3] |
P. Cannarsa and C. Sinestrari, "Semiconcave Functions, Hamiton-Jacobi Equations and Optimal Control," Birkhäser, Boston, 2004. |
[4] |
A. Canino, On $p$-convex sets and geodesics, J. Differential Equations, 75 (1988), 118-157.
doi: doi:10.1016/0022-0396(88)90132-5. |
[5] |
F. H. Clarke, "Optimization and Nonsmooth Analysis," Classics in Applied Mathematics, 5, SIAM, Philadelphia, PA, 1990. |
[6] |
F. H. Clarke, Yu. Ledyaev, R. J. Stern and P. R. Wolenski, "Nonsmooth Analysis and Control Theory," Graduate Texts in Mathematics, 178, Springer-Verlag, New York, 1998. |
[7] |
F. H. Clarke and R. J. Stern, State constrained feedback stabilization, SIAM J. Control Optim., 42 (2003), 422-441.
doi: doi:10.1137/S036301290240453X. |
[8] |
F. H. Clarke, R. J. Stern and P. R. Wolenski, Proximal smoothness and the lower-$C^2$ property, J. Convex Anal., 2 (1995), 117-144. |
[9] |
G. Colombo and A. Marigonda, Differentiability properties for a class of non-convex functions, Calc. Var., 25 (2005), 1-31.
doi: doi:10.1007/s00526-005-0352-7. |
[10] |
G. Colombo, A. Marigonda and P. R. Wolenski, Some new regularity properties for the minimal time function, SIAM J. Control Optim., 44 (2006), 2285-2299.
doi: doi:10.1137/050630076. |
[11] |
H. Federer, Curvature measures, Trans. Amer. Math. Soc., 93 (1959), 418-491. |
[12] |
C. Nour, R. J. Stern and J. Takche, Proximal smoothness and the exterior sphere condition, J. Convex Anal., 16 (2009), 501-514. |
[13] |
C. Nour, R. J. Stern and J. Takche, The $\theta$-exterior sphere condition, $\varphi$-convexity, and local semiconcavity, Nonlinear Anal. Theor. Meth. Appl., 73 (2010), 573-589.
doi: doi:10.1016/j.na.2010.04.001. |
[14] |
R. A. Poliquin and R. T. Rockafellar, Prox-regular functions in variational analysis, Trans. Amer. Math. Soc., 348 (1996), 1805-1838.
doi: doi:10.1090/S0002-9947-96-01544-9. |
[15] |
R. A. Poliquin, R. T. Rockafellar and L. Thibault, Local differentiability of distance functions, Trans. Amer. Math. Soc., 352 (2000), 5231-5249.
doi: doi:10.1090/S0002-9947-00-02550-2. |
[16] |
R. T. Rockafellar, Clarke's tangent cones and the boundaries of closed sets in $\R^n$, Nonlinear Anal. Theor. Meth. Appl., 3 (1979), 145-154.
doi: doi:10.1016/0362-546X(79)90044-0. |
[17] |
R. T. Rockafellar and R. J.-B. Wets, "Variational Analysis," Grundlehren der Mathematischen Wissenschaften, 317, Springer-Verlag, Berlin, 1998. |
[18] |
A. S. Shapiro, Existence and differentiability of metric projections in Hilbert spaces, SIAM J. Optim., 4 (1994), 231-259.
doi: doi:10.1137/0804006. |
[19] |
C. Sinestrari, Semiconcavity of the value function for exit time problems with nonsmooth target, Commun. Pure Appl. Anal., 3 (2004), 757-774.
doi: doi:10.3934/cpaa.2004.3.757. |
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